Internal problem ID [10887]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-3 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 63.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x y^{\prime \prime }+a y^{\prime }+b x y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 39
dsolve(x*diff(y(x),x$2)+a*diff(y(x),x)+b*x*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\operatorname {BesselJ}\left (\frac {a}{2}-\frac {1}{2}, \sqrt {b}\, x \right ) c_{1} +\operatorname {BesselY}\left (\frac {a}{2}-\frac {1}{2}, \sqrt {b}\, x \right ) c_{2} \right ) x^{-\frac {a}{2}+\frac {1}{2}} \]
✓ Solution by Mathematica
Time used: 0.057 (sec). Leaf size: 54
DSolve[x*y''[x]+a*y'[x]+b*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^{\frac {1}{2}-\frac {a}{2}} \left (c_1 \operatorname {BesselJ}\left (\frac {a-1}{2},\sqrt {b} x\right )+c_2 \operatorname {BesselY}\left (\frac {a-1}{2},\sqrt {b} x\right )\right ) \]